One card is lost from a well-shuffled pack of 52 playing cards. From the remaining 51 cards, a card is drawn at random and is found to be a heart. The probability that the lost card is a heart is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{4}{17}\)
Let \(E_1\) be the event that the lost card is a heart and \(E_2\) the event that it is not a heart, and let H be the event that the drawn card is a heart.
\( \displaystyle P(E_1) = \frac{13}{52} = \frac{1}{4},\quad P(E_2) = \frac{3}{4} \)
If a heart is lost, 12 hearts are left among 51 cards; otherwise all 13 hearts are left:
\( \displaystyle P(H|E_1) = \frac{12}{51},\quad P(H|E_2) = \frac{13}{51} \)
By Bayes’ theorem:
\( \displaystyle P(E_1|H) = \frac{\frac{1}{4}\times\frac{12}{51}}{\frac{1}{4}\times\frac{12}{51}+\frac{3}{4}\times\frac{13}{51}} \)
\( \displaystyle = \frac{12}{12+39} = \frac{12}{51} = \frac{4}{17} \)
Why the other options are incorrect
Option (2) \(\displaystyle\frac{1}{4}\) — Incorrect. This is \(P(E_1)\), the chance before the drawn card is seen.
Option (3) \(\displaystyle\frac{13}{51}\) — Incorrect. This is \(P(H|E_2)\), the chance of drawing a heart when the lost card is not a heart.
Option (4) \(\displaystyle\frac{13}{17}\) — Incorrect. This is \(P(E_2|H) = \frac{39}{51}\), the probability that the lost card is not a heart.